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<a href="_negative_gaussian_process_evidence_8h.html">Go to the documentation of this file.</a><div class="fragment"><div class="line"><a id="l00001" name="l00001"></a><span class="lineno">    1</span><span class="comment">//===========================================================================</span><span class="comment"></span></div>
<div class="line"><a id="l00002" name="l00002"></a><span class="lineno">    2</span><span class="comment">/*!</span></div>
<div class="line"><a id="l00003" name="l00003"></a><span class="lineno">    3</span><span class="comment"> * </span></div>
<div class="line"><a id="l00004" name="l00004"></a><span class="lineno">    4</span><span class="comment"> *</span></div>
<div class="line"><a id="l00005" name="l00005"></a><span class="lineno">    5</span><span class="comment"> * \brief       Evidence for model selection of a regularization network/Gaussian process.</span></div>
<div class="line"><a id="l00006" name="l00006"></a><span class="lineno">    6</span><span class="comment"></span> </div>
<div class="line"><a id="l00007" name="l00007"></a><span class="lineno">    7</span><span class="comment"></span> </div>
<div class="line"><a id="l00008" name="l00008"></a><span class="lineno">    8</span><span class="comment"> * </span></div>
<div class="line"><a id="l00009" name="l00009"></a><span class="lineno">    9</span><span class="comment"> *</span></div>
<div class="line"><a id="l00010" name="l00010"></a><span class="lineno">   10</span><span class="comment"> * \author      C. Igel, T. Glasmachers, O. Krause</span></div>
<div class="line"><a id="l00011" name="l00011"></a><span class="lineno">   11</span><span class="comment"> * \date        2007-2012</span></div>
<div class="line"><a id="l00012" name="l00012"></a><span class="lineno">   12</span><span class="comment"> *</span></div>
<div class="line"><a id="l00013" name="l00013"></a><span class="lineno">   13</span><span class="comment"> *</span></div>
<div class="line"><a id="l00014" name="l00014"></a><span class="lineno">   14</span><span class="comment"> * \par Copyright 1995-2017 Shark Development Team</span></div>
<div class="line"><a id="l00015" name="l00015"></a><span class="lineno">   15</span><span class="comment"> * </span></div>
<div class="line"><a id="l00016" name="l00016"></a><span class="lineno">   16</span><span class="comment"> * &lt;BR&gt;&lt;HR&gt;</span></div>
<div class="line"><a id="l00017" name="l00017"></a><span class="lineno">   17</span><span class="comment"> * This file is part of Shark.</span></div>
<div class="line"><a id="l00018" name="l00018"></a><span class="lineno">   18</span><span class="comment"> * &lt;https://shark-ml.github.io/Shark/&gt;</span></div>
<div class="line"><a id="l00019" name="l00019"></a><span class="lineno">   19</span><span class="comment"> * </span></div>
<div class="line"><a id="l00020" name="l00020"></a><span class="lineno">   20</span><span class="comment"> * Shark is free software: you can redistribute it and/or modify</span></div>
<div class="line"><a id="l00021" name="l00021"></a><span class="lineno">   21</span><span class="comment"> * it under the terms of the GNU Lesser General Public License as published </span></div>
<div class="line"><a id="l00022" name="l00022"></a><span class="lineno">   22</span><span class="comment"> * by the Free Software Foundation, either version 3 of the License, or</span></div>
<div class="line"><a id="l00023" name="l00023"></a><span class="lineno">   23</span><span class="comment"> * (at your option) any later version.</span></div>
<div class="line"><a id="l00024" name="l00024"></a><span class="lineno">   24</span><span class="comment"> * </span></div>
<div class="line"><a id="l00025" name="l00025"></a><span class="lineno">   25</span><span class="comment"> * Shark is distributed in the hope that it will be useful,</span></div>
<div class="line"><a id="l00026" name="l00026"></a><span class="lineno">   26</span><span class="comment"> * but WITHOUT ANY WARRANTY; without even the implied warranty of</span></div>
<div class="line"><a id="l00027" name="l00027"></a><span class="lineno">   27</span><span class="comment"> * MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the</span></div>
<div class="line"><a id="l00028" name="l00028"></a><span class="lineno">   28</span><span class="comment"> * GNU Lesser General Public License for more details.</span></div>
<div class="line"><a id="l00029" name="l00029"></a><span class="lineno">   29</span><span class="comment"> * </span></div>
<div class="line"><a id="l00030" name="l00030"></a><span class="lineno">   30</span><span class="comment"> * You should have received a copy of the GNU Lesser General Public License</span></div>
<div class="line"><a id="l00031" name="l00031"></a><span class="lineno">   31</span><span class="comment"> * along with Shark.  If not, see &lt;http://www.gnu.org/licenses/&gt;.</span></div>
<div class="line"><a id="l00032" name="l00032"></a><span class="lineno">   32</span><span class="comment"> *</span></div>
<div class="line"><a id="l00033" name="l00033"></a><span class="lineno">   33</span><span class="comment"> */</span></div>
<div class="line"><a id="l00034" name="l00034"></a><span class="lineno">   34</span><span class="comment">//===========================================================================</span></div>
<div class="line"><a id="l00035" name="l00035"></a><span class="lineno">   35</span> </div>
<div class="line"><a id="l00036" name="l00036"></a><span class="lineno">   36</span><span class="preprocessor">#ifndef SHARK_OBJECTIVEFUNCTIONS_NEGATIVEGAUSSIANPROCESSEVIDENCE_H</span></div>
<div class="line"><a id="l00037" name="l00037"></a><span class="lineno">   37</span><span class="preprocessor">#define SHARK_OBJECTIVEFUNCTIONS_NEGATIVEGAUSSIANPROCESSEVIDENCE_H</span></div>
<div class="line"><a id="l00038" name="l00038"></a><span class="lineno">   38</span> </div>
<div class="line"><a id="l00039" name="l00039"></a><span class="lineno">   39</span><span class="preprocessor">#include &lt;<a class="code" href="_abstract_objective_function_8h.html" title="AbstractObjectiveFunction.">shark/ObjectiveFunctions/AbstractObjectiveFunction.h</a>&gt;</span></div>
<div class="line"><a id="l00040" name="l00040"></a><span class="lineno">   40</span><span class="preprocessor">#include &lt;<a class="code" href="_kernel_helpers_8h.html">shark/Models/Kernels/KernelHelpers.h</a>&gt;</span></div>
<div class="line"><a id="l00041" name="l00041"></a><span class="lineno">   41</span> </div>
<div class="line"><a id="l00042" name="l00042"></a><span class="lineno">   42</span><span class="preprocessor">#include &lt;<a class="code" href="_base_8h.html">shark/LinAlg/Base.h</a>&gt;</span></div>
<div class="line"><a id="l00043" name="l00043"></a><span class="lineno">   43</span><span class="keyword">namespace </span><a class="code hl_namespace" href="namespaceshark.html" title="AbstractMultiObjectiveOptimizer.">shark</a> {</div>
<div class="line"><a id="l00044" name="l00044"></a><span class="lineno">   44</span> </div>
<div class="line"><a id="l00045" name="l00045"></a><span class="lineno">   45</span><span class="comment"></span> </div>
<div class="line"><a id="l00046" name="l00046"></a><span class="lineno">   46</span><span class="comment">///</span></div>
<div class="line"><a id="l00047" name="l00047"></a><span class="lineno">   47</span><span class="comment">/// \brief Evidence for model selection of a regularization network/Gaussian process.</span></div>
<div class="line"><a id="l00048" name="l00048"></a><span class="lineno">   48</span><span class="comment">///</span></div>
<div class="line"><a id="l00049" name="l00049"></a><span class="lineno">   49</span><span class="comment">/// Let \f$M\f$ denote the (kernel Gram) covariance matrix and</span></div>
<div class="line"><a id="l00050" name="l00050"></a><span class="lineno">   50</span><span class="comment">/// \f$t\f$ the corresponding label vector.  For the evidence we have: </span></div>
<div class="line"><a id="l00051" name="l00051"></a><span class="lineno">   51</span><span class="comment">/// \f[ E = 1/2 \cdot [ -\log(\det(M)) - t^T M^{-1} t - N \log(2 \pi)] \f]</span></div>
<div class="line"><a id="l00052" name="l00052"></a><span class="lineno">   52</span><span class="comment">///</span></div>
<div class="line"><a id="l00053" name="l00053"></a><span class="lineno">   53</span><span class="comment">/// The evidence is also known as marginal (log)likelihood. For</span></div>
<div class="line"><a id="l00054" name="l00054"></a><span class="lineno">   54</span><span class="comment">/// details, please see:</span></div>
<div class="line"><a id="l00055" name="l00055"></a><span class="lineno">   55</span><span class="comment">///</span></div>
<div class="line"><a id="l00056" name="l00056"></a><span class="lineno">   56</span><span class="comment">/// C.E. Rasmussen &amp; C.K.I. Williams, Gaussian</span></div>
<div class="line"><a id="l00057" name="l00057"></a><span class="lineno">   57</span><span class="comment">/// Processes for Machine Learning, section 5.4, MIT Press, 2006</span></div>
<div class="line"><a id="l00058" name="l00058"></a><span class="lineno">   58</span><span class="comment">///</span></div>
<div class="line"><a id="l00059" name="l00059"></a><span class="lineno">   59</span><span class="comment">/// C.M. Bishop, Pattern Recognition and Machine Learning, section</span></div>
<div class="line"><a id="l00060" name="l00060"></a><span class="lineno">   60</span><span class="comment">/// 6.4.3, Springer, 2006</span></div>
<div class="line"><a id="l00061" name="l00061"></a><span class="lineno">   61</span><span class="comment">///</span></div>
<div class="line"><a id="l00062" name="l00062"></a><span class="lineno">   62</span><span class="comment">/// The regularization parameter can be encoded in different ways.</span></div>
<div class="line"><a id="l00063" name="l00063"></a><span class="lineno">   63</span><span class="comment">/// The exponential encoding is the proper choice for unconstraint optimization.</span></div>
<div class="line"><a id="l00064" name="l00064"></a><span class="lineno">   64</span><span class="comment">/// Be careful not to mix up different encodings between trainer and evidence.</span></div>
<div class="line"><a id="l00065" name="l00065"></a><span class="lineno">   65</span><span class="comment">/// \ingroup kerneloptimization</span></div>
<div class="line"><a id="l00066" name="l00066"></a><span class="lineno">   66</span><span class="comment"></span><span class="keyword">template</span>&lt;<span class="keyword">class</span> InputType = RealVector, <span class="keyword">class</span> OutputType = RealVector, <span class="keyword">class</span> LabelType = RealVector&gt;</div>
<div class="foldopen" id="foldopen00067" data-start="{" data-end="};">
<div class="line"><a id="l00067" name="l00067"></a><span class="lineno"><a class="line" href="classshark_1_1_negative_gaussian_process_evidence.html">   67</a></span><span class="keyword">class </span><a class="code hl_class" href="classshark_1_1_negative_gaussian_process_evidence.html" title="Evidence for model selection of a regularization network/Gaussian process.">NegativeGaussianProcessEvidence</a> : <span class="keyword">public</span> <a class="code hl_class" href="classshark_1_1_abstract_objective_function.html" title="Super class of all objective functions for optimization and learning.">AbstractObjectiveFunction</a>&lt; RealVector, double &gt;</div>
<div class="line"><a id="l00068" name="l00068"></a><span class="lineno">   68</span>{</div>
<div class="line"><a id="l00069" name="l00069"></a><span class="lineno">   69</span><span class="keyword">public</span>:</div>
<div class="line"><a id="l00070" name="l00070"></a><span class="lineno"><a class="line" href="classshark_1_1_negative_gaussian_process_evidence.html#adaae9bffb2de9f7695fd52035fa9a794">   70</a></span>    <span class="keyword">typedef</span> <a class="code hl_class" href="classshark_1_1_labeled_data.html" title="Data set for supervised learning.">LabeledData&lt;InputType,LabelType&gt;</a> <a class="code hl_typedef" href="classshark_1_1_negative_gaussian_process_evidence.html#adaae9bffb2de9f7695fd52035fa9a794">DatasetType</a>;</div>
<div class="line"><a id="l00071" name="l00071"></a><span class="lineno"><a class="line" href="classshark_1_1_negative_gaussian_process_evidence.html#ac13877e83391d9ee87e893688ac8a109">   71</a></span>    <span class="keyword">typedef</span> <a class="code hl_class" href="classshark_1_1_abstract_kernel_function.html" title="Base class of all Kernel functions.">AbstractKernelFunction&lt;InputType&gt;</a> <a class="code hl_typedef" href="classshark_1_1_negative_gaussian_process_evidence.html#ac13877e83391d9ee87e893688ac8a109">KernelType</a>;</div>
<div class="line"><a id="l00072" name="l00072"></a><span class="lineno">   72</span><span class="comment"></span> </div>
<div class="line"><a id="l00073" name="l00073"></a><span class="lineno">   73</span><span class="comment">    /// \param dataset: training data for the Gaussian process</span></div>
<div class="line"><a id="l00074" name="l00074"></a><span class="lineno">   74</span><span class="comment">    /// \param kernel: pointer to external kernel function</span></div>
<div class="line"><a id="l00075" name="l00075"></a><span class="lineno">   75</span><span class="comment">    /// \param unconstrained: exponential encoding of regularization parameter for unconstraint optimization</span></div>
<div class="foldopen" id="foldopen00076" data-start="{" data-end="}">
<div class="line"><a id="l00076" name="l00076"></a><span class="lineno"><a class="line" href="classshark_1_1_negative_gaussian_process_evidence.html#a3de0372ab79585993728bfd7dbb60438">   76</a></span><span class="comment"></span>    <a class="code hl_function" href="classshark_1_1_negative_gaussian_process_evidence.html#a3de0372ab79585993728bfd7dbb60438">NegativeGaussianProcessEvidence</a>(</div>
<div class="line"><a id="l00077" name="l00077"></a><span class="lineno">   77</span>        <a class="code hl_class" href="classshark_1_1_labeled_data.html">DatasetType</a> <span class="keyword">const</span>&amp; dataset,</div>
<div class="line"><a id="l00078" name="l00078"></a><span class="lineno">   78</span>        <a class="code hl_class" href="classshark_1_1_abstract_kernel_function.html">KernelType</a>* kernel,</div>
<div class="line"><a id="l00079" name="l00079"></a><span class="lineno">   79</span>        <span class="keywordtype">bool</span> unconstrained = <span class="keyword">false</span></div>
<div class="line"><a id="l00080" name="l00080"></a><span class="lineno">   80</span>    ): m_dataset(dataset)</div>
<div class="line"><a id="l00081" name="l00081"></a><span class="lineno">   81</span>    , mep_kernel(kernel)</div>
<div class="line"><a id="l00082" name="l00082"></a><span class="lineno">   82</span>    , m_unconstrained(unconstrained)</div>
<div class="line"><a id="l00083" name="l00083"></a><span class="lineno">   83</span>    {</div>
<div class="line"><a id="l00084" name="l00084"></a><span class="lineno">   84</span>        <span class="keywordflow">if</span> (kernel-&gt;<a class="code hl_function" href="classshark_1_1_abstract_kernel_function.html#ac0c799ac75db64200256ed50d34d2411">hasFirstParameterDerivative</a>()) <a class="code hl_variable" href="classshark_1_1_abstract_objective_function.html#ad8888c58fd3f98e73013afb5dd4b2af1">m_features</a> |= <a class="code hl_enumvalue" href="classshark_1_1_abstract_objective_function.html#aadafeb6dfb5b649f321e7b81ac8aad1aa0bc7673a369df5f86ddd6ba6735f4971" title="The method evalDerivative is implemented for the first derivative and returns a sensible value.">HAS_FIRST_DERIVATIVE</a>;</div>
<div class="line"><a id="l00085" name="l00085"></a><span class="lineno">   85</span>        <a class="code hl_function" href="classshark_1_1_negative_gaussian_process_evidence.html#a73e8d2e496a680a894266fadb2c554e0" title="set threshold value for truncating partial derivatives">setThreshold</a>(0.);</div>
<div class="line"><a id="l00086" name="l00086"></a><span class="lineno">   86</span>    }</div>
</div>
<div class="line"><a id="l00087" name="l00087"></a><span class="lineno">   87</span><span class="comment"></span> </div>
<div class="line"><a id="l00088" name="l00088"></a><span class="lineno">   88</span><span class="comment">    /// \brief From INameable: return the class name.</span></div>
<div class="foldopen" id="foldopen00089" data-start="{" data-end="}">
<div class="line"><a id="l00089" name="l00089"></a><span class="lineno"><a class="line" href="classshark_1_1_negative_gaussian_process_evidence.html#af8d8184ce532bfefe178d8189c417989">   89</a></span><span class="comment"></span>    std::string <a class="code hl_function" href="classshark_1_1_negative_gaussian_process_evidence.html#af8d8184ce532bfefe178d8189c417989" title="From INameable: return the class name.">name</a>()<span class="keyword"> const</span></div>
<div class="line"><a id="l00090" name="l00090"></a><span class="lineno">   90</span><span class="keyword">    </span>{ <span class="keywordflow">return</span> <span class="stringliteral">&quot;NegativeGaussianProcessEvidence&quot;</span>; }</div>
</div>
<div class="line"><a id="l00091" name="l00091"></a><span class="lineno">   91</span>    </div>
<div class="foldopen" id="foldopen00092" data-start="{" data-end="}">
<div class="line"><a id="l00092" name="l00092"></a><span class="lineno"><a class="line" href="classshark_1_1_negative_gaussian_process_evidence.html#a1ae734c2f62e91619ac17a437b6fd224">   92</a></span>    std::size_t <a class="code hl_function" href="classshark_1_1_negative_gaussian_process_evidence.html#a1ae734c2f62e91619ac17a437b6fd224" title="Accesses the number of variables.">numberOfVariables</a>()<span class="keyword">const</span>{</div>
<div class="line"><a id="l00093" name="l00093"></a><span class="lineno">   93</span>        <span class="keywordflow">return</span> 1+ mep_kernel-&gt;<a class="code hl_function" href="classshark_1_1_i_parameterizable.html#aed1e8d1d4dbde387e2f6a25141ed3a20" title="Return the number of parameters.">numberOfParameters</a>();</div>
<div class="line"><a id="l00094" name="l00094"></a><span class="lineno">   94</span>    }</div>
</div>
<div class="line"><a id="l00095" name="l00095"></a><span class="lineno">   95</span><span class="comment"></span> </div>
<div class="line"><a id="l00096" name="l00096"></a><span class="lineno">   96</span><span class="comment">    /// Let \f$M\f$ denote the (kernel Gram) covariance matrix and</span></div>
<div class="line"><a id="l00097" name="l00097"></a><span class="lineno">   97</span><span class="comment">    /// \f$t\f$ the label vector.  For the evidence we have: \f[ E= 1/2 \cdot [ -\log(\det(M)) - t^T M^{-1} t - N \log(2 \pi) ] \f]</span></div>
<div class="foldopen" id="foldopen00098" data-start="{" data-end="}">
<div class="line"><a id="l00098" name="l00098"></a><span class="lineno"><a class="line" href="classshark_1_1_negative_gaussian_process_evidence.html#a9c74a1a22f2496b879cc1683ee15bc86">   98</a></span><span class="comment"></span>    <span class="keywordtype">double</span> <a class="code hl_function" href="classshark_1_1_negative_gaussian_process_evidence.html#a9c74a1a22f2496b879cc1683ee15bc86">eval</a>(<span class="keyword">const</span> RealVector&amp; parameters)<span class="keyword"> const </span>{</div>
<div class="line"><a id="l00099" name="l00099"></a><span class="lineno">   99</span>        std::size_t N  = m_dataset.<a class="code hl_function" href="group__shark__globals.html#ga5333445992cd6b14392cd80a1ab5403c" title="Returns the total number of elements.">numberOfElements</a>(); </div>
<div class="line"><a id="l00100" name="l00100"></a><span class="lineno">  100</span>        std::size_t kp = mep_kernel-&gt;<a class="code hl_function" href="classshark_1_1_i_parameterizable.html#aed1e8d1d4dbde387e2f6a25141ed3a20" title="Return the number of parameters.">numberOfParameters</a>();</div>
<div class="line"><a id="l00101" name="l00101"></a><span class="lineno">  101</span>        <span class="comment">// check whether argument has right dimensionality</span></div>
<div class="line"><a id="l00102" name="l00102"></a><span class="lineno">  102</span>        <a class="code hl_define" href="_exception_8h.html#a73abb5049a0168d72a48e72dda41708b">SHARK_ASSERT</a>(1+kp == parameters.size());</div>
<div class="line"><a id="l00103" name="l00103"></a><span class="lineno">  103</span> </div>
<div class="line"><a id="l00104" name="l00104"></a><span class="lineno">  104</span>        <span class="comment">// keep track of how often the objective function is called</span></div>
<div class="line"><a id="l00105" name="l00105"></a><span class="lineno">  105</span>        <a class="code hl_variable" href="classshark_1_1_abstract_objective_function.html#af0942c072be06d0dd4da5ee7067c5777" title="Evaluation counter, default value: 0.">m_evaluationCounter</a>++;</div>
<div class="line"><a id="l00106" name="l00106"></a><span class="lineno">  106</span>        </div>
<div class="line"><a id="l00107" name="l00107"></a><span class="lineno">  107</span>        <span class="comment">//set parameters</span></div>
<div class="line"><a id="l00108" name="l00108"></a><span class="lineno">  108</span>        <span class="keywordtype">double</span> betaInv = parameters.back();</div>
<div class="line"><a id="l00109" name="l00109"></a><span class="lineno">  109</span>        <span class="keywordflow">if</span>(m_unconstrained)</div>
<div class="line"><a id="l00110" name="l00110"></a><span class="lineno">  110</span>            betaInv = std::exp(betaInv); <span class="comment">// for unconstraint optimization</span></div>
<div class="line"><a id="l00111" name="l00111"></a><span class="lineno">  111</span>        mep_kernel-&gt;<a class="code hl_function" href="classshark_1_1_i_parameterizable.html#ad5e35d1a10ff36fa72ea787baa40e9ad" title="Set the parameter vector.">setParameterVector</a>(subrange(parameters,0,kp));</div>
<div class="line"><a id="l00112" name="l00112"></a><span class="lineno">  112</span>        </div>
<div class="line"><a id="l00113" name="l00113"></a><span class="lineno">  113</span>        </div>
<div class="line"><a id="l00114" name="l00114"></a><span class="lineno">  114</span>        <span class="comment">//generate kernel matrix and label vector</span></div>
<div class="line"><a id="l00115" name="l00115"></a><span class="lineno">  115</span>        RealMatrix M = <a class="code hl_function" href="group__kernels.html#ga3eaca71bfc1467b79c9341dcfcad25c1" title="Calculates the regularized kernel gram matrix of the points stored inside a dataset.">calculateRegularizedKernelMatrix</a>(*mep_kernel,m_dataset.<a class="code hl_function" href="group__shark__globals.html#ga6f74e657c7e0c8a32b2456fb328bd653" title="Access to inputs as a separate container.">inputs</a>(),betaInv);</div>
<div class="line"><a id="l00116" name="l00116"></a><span class="lineno">  116</span>        RealMatrix t = createBatch&lt;RealVector&gt;(m_dataset.<a class="code hl_function" href="group__shark__globals.html#ga6328a5aa2570c01a5ac5f25076071663" title="Access to labels as a separate container.">labels</a>().<a class="code hl_function" href="group__shark__globals.html#gad9b0233e3adc882ed94f418f80767b09" title="Returns the range of elements.">elements</a>());</div>
<div class="line"><a id="l00117" name="l00117"></a><span class="lineno">  117</span> </div>
<div class="line"><a id="l00118" name="l00118"></a><span class="lineno">  118</span>        blas::cholesky_decomposition&lt;RealMatrix&gt; cholesky(M);</div>
<div class="line"><a id="l00119" name="l00119"></a><span class="lineno">  119</span>        </div>
<div class="line"><a id="l00120" name="l00120"></a><span class="lineno">  120</span>        <span class="comment">//compute the determinant of M using the cholesky factorization M=AA^T:</span></div>
<div class="line"><a id="l00121" name="l00121"></a><span class="lineno">  121</span>        <span class="comment">//ln det(M) = 2 trace(ln A)</span></div>
<div class="line"><a id="l00122" name="l00122"></a><span class="lineno">  122</span>        <span class="keywordtype">double</span> logDet = 2* trace(log(cholesky.lower_factor()));</div>
<div class="line"><a id="l00123" name="l00123"></a><span class="lineno">  123</span>        </div>
<div class="line"><a id="l00124" name="l00124"></a><span class="lineno">  124</span>        <span class="comment">//we need to compute t^T M^-1 t </span></div>
<div class="line"><a id="l00125" name="l00125"></a><span class="lineno">  125</span>        <span class="comment">//= t^T (AA^T)^-1 t= t^T (A^-T A^-1)=||A^-1 t||^2</span></div>
<div class="line"><a id="l00126" name="l00126"></a><span class="lineno">  126</span>        <span class="comment">//so we will first solve the triangular System Az=t</span></div>
<div class="line"><a id="l00127" name="l00127"></a><span class="lineno">  127</span>        <span class="comment">//and then compute ||z||^2</span></div>
<div class="line"><a id="l00128" name="l00128"></a><span class="lineno">  128</span>        RealMatrix z = solve(cholesky.lower_factor(),t,blas::lower(), blas::left());</div>
<div class="line"><a id="l00129" name="l00129"></a><span class="lineno">  129</span> </div>
<div class="line"><a id="l00130" name="l00130"></a><span class="lineno">  130</span>        <span class="comment">// equation (6.69) on page 311 in the book C.M. Bishop, Pattern Recognition and Machine Learning, Springer, 2006</span></div>
<div class="line"><a id="l00131" name="l00131"></a><span class="lineno">  131</span>        <span class="comment">// e = 1/2 \cdot [ -log(det(M)) - t^T M^{-1} t - N log(2 \pi) ]</span></div>
<div class="line"><a id="l00132" name="l00132"></a><span class="lineno">  132</span>        <span class="keywordtype">double</span> e = 0.5 * (-logDet - norm_sqr(to_vector(z)) - N * std::log(2.0 * M_PI));</div>
<div class="line"><a id="l00133" name="l00133"></a><span class="lineno">  133</span> </div>
<div class="line"><a id="l00134" name="l00134"></a><span class="lineno">  134</span>        <span class="comment">// return the *negative* evidence</span></div>
<div class="line"><a id="l00135" name="l00135"></a><span class="lineno">  135</span>        <span class="keywordflow">return</span> -e;</div>
<div class="line"><a id="l00136" name="l00136"></a><span class="lineno">  136</span>    }</div>
</div>
<div class="line"><a id="l00137" name="l00137"></a><span class="lineno">  137</span><span class="comment"></span> </div>
<div class="line"><a id="l00138" name="l00138"></a><span class="lineno">  138</span><span class="comment">    /// Let \f$M\f$ denote the regularized (kernel Gram) covariance matrix.</span></div>
<div class="line"><a id="l00139" name="l00139"></a><span class="lineno">  139</span><span class="comment">    /// For the evidence we have:</span></div>
<div class="line"><a id="l00140" name="l00140"></a><span class="lineno">  140</span><span class="comment">    /// \f[ E = 1/2 \cdot [ -\log(\det(M)) - t^T M^{-1} t - N \log(2 \pi) ] \f]</span></div>
<div class="line"><a id="l00141" name="l00141"></a><span class="lineno">  141</span><span class="comment">    /// For a kernel parameter \f$p\f$ and \f$C = \beta^{-1}\f$ we get the derivatives:</span></div>
<div class="line"><a id="l00142" name="l00142"></a><span class="lineno">  142</span><span class="comment">    /// \f[  dE/dC = 1/2 \cdot [ -tr(M^{-1}) + (M^{-1} t)^2 ] \f]</span></div>
<div class="line"><a id="l00143" name="l00143"></a><span class="lineno">  143</span><span class="comment">    /// \f[  dE/dp = 1/2 \cdot [ -tr(M^{-1} dM/dp) + t^T (M^{-1} dM/dp M^{-1}) t ] \f]</span></div>
<div class="foldopen" id="foldopen00144" data-start="{" data-end="}">
<div class="line"><a id="l00144" name="l00144"></a><span class="lineno"><a class="line" href="classshark_1_1_negative_gaussian_process_evidence.html#a4bbd89a9d2c47ecc601fb23567715b0d">  144</a></span><span class="comment"></span>    <span class="keywordtype">double</span> <a class="code hl_function" href="classshark_1_1_negative_gaussian_process_evidence.html#a4bbd89a9d2c47ecc601fb23567715b0d">evalDerivative</a>(<span class="keyword">const</span> RealVector&amp; parameters, <a class="code hl_typedef" href="classshark_1_1_abstract_objective_function.html#a29804371954a360f09696adea7cfd839">FirstOrderDerivative</a>&amp; derivative)<span class="keyword"> const </span>{</div>
<div class="line"><a id="l00145" name="l00145"></a><span class="lineno">  145</span>        std::size_t N  = m_dataset.<a class="code hl_function" href="group__shark__globals.html#ga5333445992cd6b14392cd80a1ab5403c" title="Returns the total number of elements.">numberOfElements</a>(); </div>
<div class="line"><a id="l00146" name="l00146"></a><span class="lineno">  146</span>        std::size_t kp = mep_kernel-&gt;<a class="code hl_function" href="classshark_1_1_i_parameterizable.html#aed1e8d1d4dbde387e2f6a25141ed3a20" title="Return the number of parameters.">numberOfParameters</a>();</div>
<div class="line"><a id="l00147" name="l00147"></a><span class="lineno">  147</span> </div>
<div class="line"><a id="l00148" name="l00148"></a><span class="lineno">  148</span>        <span class="comment">// check whether argument has right dimensionality</span></div>
<div class="line"><a id="l00149" name="l00149"></a><span class="lineno">  149</span>        <a class="code hl_define" href="_exception_8h.html#a73abb5049a0168d72a48e72dda41708b">SHARK_ASSERT</a>(1 + kp == parameters.size());</div>
<div class="line"><a id="l00150" name="l00150"></a><span class="lineno">  150</span>        derivative.resize(1 + kp);</div>
<div class="line"><a id="l00151" name="l00151"></a><span class="lineno">  151</span>        </div>
<div class="line"><a id="l00152" name="l00152"></a><span class="lineno">  152</span>        <span class="comment">// keep track of how often the objective function is called</span></div>
<div class="line"><a id="l00153" name="l00153"></a><span class="lineno">  153</span>        <a class="code hl_variable" href="classshark_1_1_abstract_objective_function.html#af0942c072be06d0dd4da5ee7067c5777" title="Evaluation counter, default value: 0.">m_evaluationCounter</a>++;</div>
<div class="line"><a id="l00154" name="l00154"></a><span class="lineno">  154</span> </div>
<div class="line"><a id="l00155" name="l00155"></a><span class="lineno">  155</span>        <span class="comment">//set parameters</span></div>
<div class="line"><a id="l00156" name="l00156"></a><span class="lineno">  156</span>        <span class="keywordtype">double</span> betaInv = parameters.back();</div>
<div class="line"><a id="l00157" name="l00157"></a><span class="lineno">  157</span>        <span class="keywordflow">if</span>(m_unconstrained)</div>
<div class="line"><a id="l00158" name="l00158"></a><span class="lineno">  158</span>            betaInv = std::exp(betaInv); <span class="comment">// for unconstraint optimization</span></div>
<div class="line"><a id="l00159" name="l00159"></a><span class="lineno">  159</span>        mep_kernel-&gt;<a class="code hl_function" href="classshark_1_1_i_parameterizable.html#ad5e35d1a10ff36fa72ea787baa40e9ad" title="Set the parameter vector.">setParameterVector</a>(subrange(parameters,0,kp));</div>
<div class="line"><a id="l00160" name="l00160"></a><span class="lineno">  160</span>        </div>
<div class="line"><a id="l00161" name="l00161"></a><span class="lineno">  161</span>        <span class="comment">//generate kernel matrix and label vector</span></div>
<div class="line"><a id="l00162" name="l00162"></a><span class="lineno">  162</span>        RealMatrix M = <a class="code hl_function" href="group__kernels.html#ga3eaca71bfc1467b79c9341dcfcad25c1" title="Calculates the regularized kernel gram matrix of the points stored inside a dataset.">calculateRegularizedKernelMatrix</a>(*mep_kernel,m_dataset.<a class="code hl_function" href="group__shark__globals.html#ga6f74e657c7e0c8a32b2456fb328bd653" title="Access to inputs as a separate container.">inputs</a>(),betaInv);</div>
<div class="line"><a id="l00163" name="l00163"></a><span class="lineno">  163</span>        RealMatrix t = createBatch&lt;RealVector&gt;(m_dataset.<a class="code hl_function" href="group__shark__globals.html#ga6328a5aa2570c01a5ac5f25076071663" title="Access to labels as a separate container.">labels</a>().<a class="code hl_function" href="group__shark__globals.html#gad9b0233e3adc882ed94f418f80767b09" title="Returns the range of elements.">elements</a>());</div>
<div class="line"><a id="l00164" name="l00164"></a><span class="lineno">  164</span> </div>
<div class="line"><a id="l00165" name="l00165"></a><span class="lineno">  165</span>        <span class="comment">//compute cholesky decomposition of M</span></div>
<div class="line"><a id="l00166" name="l00166"></a><span class="lineno">  166</span>        blas::cholesky_decomposition&lt;RealMatrix&gt; cholesky(M);</div>
<div class="line"><a id="l00167" name="l00167"></a><span class="lineno">  167</span>        <span class="comment">//we dont need M anymore, so save a lot of memory by freeing the memory of M</span></div>
<div class="line"><a id="l00168" name="l00168"></a><span class="lineno">  168</span>        M=RealMatrix();</div>
<div class="line"><a id="l00169" name="l00169"></a><span class="lineno">  169</span>        </div>
<div class="line"><a id="l00170" name="l00170"></a><span class="lineno">  170</span>        <span class="comment">// compute derivative w.r.t. kernel parameters</span></div>
<div class="line"><a id="l00171" name="l00171"></a><span class="lineno">  171</span>        <span class="comment">//the derivative is defined as:</span></div>
<div class="line"><a id="l00172" name="l00172"></a><span class="lineno">  172</span>        <span class="comment">//dE/da = -tr(IM dM/da) +t^T IM dM/da IM t</span></div>
<div class="line"><a id="l00173" name="l00173"></a><span class="lineno">  173</span>        <span class="comment">// where IM is the inverse matrix of M, tr is the trace and a are the parameters of the kernel</span></div>
<div class="line"><a id="l00174" name="l00174"></a><span class="lineno">  174</span>        <span class="comment">//by substituting z = IM t we can expand the operations to:</span></div>
<div class="line"><a id="l00175" name="l00175"></a><span class="lineno">  175</span>        <span class="comment">//dE/da = -(sum_i sum_j IM_ij * dM_ji/da)+(sum_i sum_j dM_ij/da *z_i * z_j)</span></div>
<div class="line"><a id="l00176" name="l00176"></a><span class="lineno">  176</span>        <span class="comment">//           =  sum_i sum_j (-IM_ij+z_i * z_j) * dM_ij/da</span></div>
<div class="line"><a id="l00177" name="l00177"></a><span class="lineno">  177</span>        <span class="comment">// with W = -IM + zz^T we get</span></div>
<div class="line"><a id="l00178" name="l00178"></a><span class="lineno">  178</span>        <span class="comment">// dE/da = sum_i sum_j W dM_ij/da</span></div>
<div class="line"><a id="l00179" name="l00179"></a><span class="lineno">  179</span>        <span class="comment">//this can be calculated as blockwise derivative.</span></div>
<div class="line"><a id="l00180" name="l00180"></a><span class="lineno">  180</span>        </div>
<div class="line"><a id="l00181" name="l00181"></a><span class="lineno">  181</span>        <span class="comment">//compute inverse matrix from the cholesky decomposition </span></div>
<div class="line"><a id="l00182" name="l00182"></a><span class="lineno">  182</span>        RealMatrix W= blas::identity_matrix&lt;double&gt;(N);</div>
<div class="line"><a id="l00183" name="l00183"></a><span class="lineno">  183</span>        cholesky.solve(W,blas::left());</div>
<div class="line"><a id="l00184" name="l00184"></a><span class="lineno">  184</span> </div>
<div class="line"><a id="l00185" name="l00185"></a><span class="lineno">  185</span>        <span class="comment">//calculate z = Wt=M^-1 t</span></div>
<div class="line"><a id="l00186" name="l00186"></a><span class="lineno">  186</span>        RealMatrix z = prod(W,t);</div>
<div class="line"><a id="l00187" name="l00187"></a><span class="lineno">  187</span>        </div>
<div class="line"><a id="l00188" name="l00188"></a><span class="lineno">  188</span>        <span class="comment">// W is already initialized as the inverse of M, so we only need </span></div>
<div class="line"><a id="l00189" name="l00189"></a><span class="lineno">  189</span>        <span class="comment">// to change the sign and add z. to calculate W fully</span></div>
<div class="line"><a id="l00190" name="l00190"></a><span class="lineno">  190</span>        W*=-1;</div>
<div class="line"><a id="l00191" name="l00191"></a><span class="lineno">  191</span>        noalias(W) += prod(z,trans(z));</div>
<div class="line"><a id="l00192" name="l00192"></a><span class="lineno">  192</span>        </div>
<div class="line"><a id="l00193" name="l00193"></a><span class="lineno">  193</span>        </div>
<div class="line"><a id="l00194" name="l00194"></a><span class="lineno">  194</span>        <span class="comment">//now calculate the derivative</span></div>
<div class="line"><a id="l00195" name="l00195"></a><span class="lineno">  195</span>        RealVector kernelGradient = 0.5*<a class="code hl_function" href="group__kernels.html#gafb6b639ff5daa090b08b13e97e78a7bc" title="Efficiently calculates the weighted derivative of a Kernel Gram Matrix w.r.t the Kernel Parameters.">calculateKernelMatrixParameterDerivative</a>(*mep_kernel,m_dataset.<a class="code hl_function" href="group__shark__globals.html#ga6f74e657c7e0c8a32b2456fb328bd653" title="Access to inputs as a separate container.">inputs</a>(),W);</div>
<div class="line"><a id="l00196" name="l00196"></a><span class="lineno">  196</span>        </div>
<div class="line"><a id="l00197" name="l00197"></a><span class="lineno">  197</span>        <span class="comment">// compute derivative w.r.t. regularization parameter</span></div>
<div class="line"><a id="l00198" name="l00198"></a><span class="lineno">  198</span>        <span class="comment">//we have: dE/dC = 1/2 * [ -tr(M^{-1}) + (M^{-1} t)^2</span></div>
<div class="line"><a id="l00199" name="l00199"></a><span class="lineno">  199</span>        <span class="comment">// which can also be written as 1/2 tr(W)</span></div>
<div class="line"><a id="l00200" name="l00200"></a><span class="lineno">  200</span>        <span class="keywordtype">double</span> betaInvDerivative = 0.5 * trace(W) ;</div>
<div class="line"><a id="l00201" name="l00201"></a><span class="lineno">  201</span>        <span class="keywordflow">if</span>(m_unconstrained) </div>
<div class="line"><a id="l00202" name="l00202"></a><span class="lineno">  202</span>            betaInvDerivative *= betaInv;</div>
<div class="line"><a id="l00203" name="l00203"></a><span class="lineno">  203</span>        </div>
<div class="line"><a id="l00204" name="l00204"></a><span class="lineno">  204</span>        <span class="comment">//merge both derivatives and since we return the negative evidence, multiply with -1</span></div>
<div class="line"><a id="l00205" name="l00205"></a><span class="lineno">  205</span>        noalias(derivative) = - (kernelGradient | betaInvDerivative);</div>
<div class="line"><a id="l00206" name="l00206"></a><span class="lineno">  206</span> </div>
<div class="line"><a id="l00207" name="l00207"></a><span class="lineno">  207</span>        <span class="comment">// truncate gradient vector </span></div>
<div class="line"><a id="l00208" name="l00208"></a><span class="lineno">  208</span>        <span class="keywordflow">for</span>(std::size_t i=0; i&lt;derivative.size(); i++) </div>
<div class="line"><a id="l00209" name="l00209"></a><span class="lineno">  209</span>            <span class="keywordflow">if</span>(std::abs(derivative(i)) &lt; m_derivativeThresholds(i)) derivative(i) = 0;</div>
<div class="line"><a id="l00210" name="l00210"></a><span class="lineno">  210</span> </div>
<div class="line"><a id="l00211" name="l00211"></a><span class="lineno">  211</span>        <span class="comment">// compute the evidence</span></div>
<div class="line"><a id="l00212" name="l00212"></a><span class="lineno">  212</span>        <span class="comment">//compute determinant of M (see eval for why this works)</span></div>
<div class="line"><a id="l00213" name="l00213"></a><span class="lineno">  213</span>        <span class="keywordtype">double</span> logDetM = 2* trace(log(cholesky.lower_factor()));</div>
<div class="line"><a id="l00214" name="l00214"></a><span class="lineno">  214</span>        <span class="keywordtype">double</span> e = 0.5 * (-logDetM - inner_prod(to_vector(t), to_vector(z)) - N * std::log(2.0 * M_PI));</div>
<div class="line"><a id="l00215" name="l00215"></a><span class="lineno">  215</span>        <span class="keywordflow">return</span> -e;</div>
<div class="line"><a id="l00216" name="l00216"></a><span class="lineno">  216</span>    }</div>
</div>
<div class="line"><a id="l00217" name="l00217"></a><span class="lineno">  217</span>    <span class="comment"></span></div>
<div class="line"><a id="l00218" name="l00218"></a><span class="lineno">  218</span><span class="comment">    /// set threshold value for truncating partial derivatives</span></div>
<div class="foldopen" id="foldopen00219" data-start="{" data-end="}">
<div class="line"><a id="l00219" name="l00219"></a><span class="lineno"><a class="line" href="classshark_1_1_negative_gaussian_process_evidence.html#a73e8d2e496a680a894266fadb2c554e0">  219</a></span><span class="comment"></span>    <span class="keywordtype">void</span> <a class="code hl_function" href="classshark_1_1_negative_gaussian_process_evidence.html#a73e8d2e496a680a894266fadb2c554e0" title="set threshold value for truncating partial derivatives">setThreshold</a>(<span class="keywordtype">double</span> d) {</div>
<div class="line"><a id="l00220" name="l00220"></a><span class="lineno">  220</span>        m_derivativeThresholds = RealVector(mep_kernel-&gt;<a class="code hl_function" href="classshark_1_1_i_parameterizable.html#aed1e8d1d4dbde387e2f6a25141ed3a20" title="Return the number of parameters.">numberOfParameters</a>() + 1, d); <span class="comment">// plus one parameter for the prior </span></div>
<div class="line"><a id="l00221" name="l00221"></a><span class="lineno">  221</span>    }</div>
</div>
<div class="line"><a id="l00222" name="l00222"></a><span class="lineno">  222</span><span class="comment"></span> </div>
<div class="line"><a id="l00223" name="l00223"></a><span class="lineno">  223</span><span class="comment">    /// set threshold values for truncating partial derivatives</span></div>
<div class="foldopen" id="foldopen00224" data-start="{" data-end="}">
<div class="line"><a id="l00224" name="l00224"></a><span class="lineno"><a class="line" href="classshark_1_1_negative_gaussian_process_evidence.html#aaf3ba1dfa906ef2dffe3ad08c5ae4bdb">  224</a></span><span class="comment"></span>    <span class="keywordtype">void</span> <a class="code hl_function" href="classshark_1_1_negative_gaussian_process_evidence.html#aaf3ba1dfa906ef2dffe3ad08c5ae4bdb" title="set threshold values for truncating partial derivatives">setThresholds</a>(RealVector &amp;c) {</div>
<div class="line"><a id="l00225" name="l00225"></a><span class="lineno">  225</span>        <a class="code hl_define" href="_exception_8h.html#a73abb5049a0168d72a48e72dda41708b">SHARK_ASSERT</a>(m_derivativeThresholds.size() == c.size());</div>
<div class="line"><a id="l00226" name="l00226"></a><span class="lineno">  226</span>        m_derivativeThresholds = c;</div>
<div class="line"><a id="l00227" name="l00227"></a><span class="lineno">  227</span>    }</div>
</div>
<div class="line"><a id="l00228" name="l00228"></a><span class="lineno">  228</span>        </div>
<div class="line"><a id="l00229" name="l00229"></a><span class="lineno">  229</span> </div>
<div class="line"><a id="l00230" name="l00230"></a><span class="lineno">  230</span><span class="keyword">private</span>:<span class="comment"></span></div>
<div class="line"><a id="l00231" name="l00231"></a><span class="lineno">  231</span><span class="comment">    /// pointer to external data set</span></div>
<div class="line"><a id="l00232" name="l00232"></a><span class="lineno">  232</span><span class="comment"></span>    <a class="code hl_typedef" href="classshark_1_1_negative_gaussian_process_evidence.html#adaae9bffb2de9f7695fd52035fa9a794">DatasetType</a> m_dataset;</div>
<div class="line"><a id="l00233" name="l00233"></a><span class="lineno">  233</span><span class="comment"></span> </div>
<div class="line"><a id="l00234" name="l00234"></a><span class="lineno">  234</span><span class="comment">    /// thresholds for setting derivatives to zero</span></div>
<div class="line"><a id="l00235" name="l00235"></a><span class="lineno">  235</span><span class="comment"></span>    RealVector  m_derivativeThresholds;</div>
<div class="line"><a id="l00236" name="l00236"></a><span class="lineno">  236</span><span class="comment"></span> </div>
<div class="line"><a id="l00237" name="l00237"></a><span class="lineno">  237</span><span class="comment">    /// pointer to external kernel function</span></div>
<div class="line"><a id="l00238" name="l00238"></a><span class="lineno">  238</span><span class="comment"></span>    <a class="code hl_typedef" href="classshark_1_1_negative_gaussian_process_evidence.html#ac13877e83391d9ee87e893688ac8a109">KernelType</a>* mep_kernel;</div>
<div class="line"><a id="l00239" name="l00239"></a><span class="lineno">  239</span><span class="comment"></span> </div>
<div class="line"><a id="l00240" name="l00240"></a><span class="lineno">  240</span><span class="comment">    /// Indicates whether log() of the regularization parameter is</span></div>
<div class="line"><a id="l00241" name="l00241"></a><span class="lineno">  241</span><span class="comment">    /// considered. This is useful for unconstraint</span></div>
<div class="line"><a id="l00242" name="l00242"></a><span class="lineno">  242</span><span class="comment">    /// optimization. The default value is false.</span></div>
<div class="line"><a id="l00243" name="l00243"></a><span class="lineno">  243</span><span class="comment"></span>    <span class="keywordtype">bool</span> m_unconstrained; </div>
<div class="line"><a id="l00244" name="l00244"></a><span class="lineno">  244</span>};</div>
</div>
<div class="line"><a id="l00245" name="l00245"></a><span class="lineno">  245</span> </div>
<div class="line"><a id="l00246" name="l00246"></a><span class="lineno">  246</span> </div>
<div class="line"><a id="l00247" name="l00247"></a><span class="lineno">  247</span>}</div>
<div class="line"><a id="l00248" name="l00248"></a><span class="lineno">  248</span><span class="preprocessor">#endif</span></div>
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